We consider singular integral operators of the form (a)$Z$_{1}L^{−1}Z_{2}, (b)$Z$_{1}Z_{2}L^{−1}, and (c)$L$^{−1}Z_{1}Z_{2}, where$Z$_{1}and$Z$_{2}are nonzero right-invariant vector fields, and$L$is the$L$^{2}-closure of a canonical Laplacian. The operators (a) are shown to be bounded on$L$^{p}for all$p$∈(1, ∞) and of weak type (1, 1), whereas all of the operators in (b) and (c) are not of weak type ($p, p$) for any$p$∈[1, ∞).

The first part of this paper describes some important underlying themes in the mathematical theory of continuum mechanics that are distinct from formulating and analyzing governing equations. The main part of this paper is devoted to a survey of some concrete, conceptually simple, pretty problems that help illuminate the underlying themes. The paper concludes with a discussion of the crucial role of invariant constitutive equations in computation.

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In this article, we classify all standard invariants that can arise from a composed inclusion of an A 3 with an A 4 subfactor. More precisely, if N P is an A 3 subfactor and P M is an A 4 subfactor, then only four standard invariants can arise from the composed inclusion N M. We answer a question posed by Bisch and Haagerup in 1994. The techniques of this paper also show that there are exactly four standard invariants for the composed inclusion of two A 4 subfactors.

Heavy Snow will turn to a natural disaster, and will bring big economic losses. The authors hope to establish a mathematic model and a plan to illustrate how to sweep off the snow on main roads in a city so as to ensure the smooth flow of traffic. The research will become a basis on which the government can workout some plans against the snow disasters. Firstly we made some assumptions, then studied the working process of snow-sweepers and derived a snow sweeping model. Based on this model, the relationship between working speed of snow-sweeper and snow thickness, a working model of snow-sweeper, and minimum sets of snow-sweeper needed were established. A deep-searching model and a model of snow-sweeping tree, and a computer program in PASCAL to determine the minimum number of snow-sweeper and the snow sweeping plans on main roads were obtained. We tried several solutions to deal with this problem, such as using piecewise curve fitting to transform a formula h~ vc relating to a cubic equation operation to formula vc~ h ,using trial and error method to determine the minimum number of snow sweepers, using deep searching and tree structure in one-stroke processing in a complicated two-way road system, and also a matrix to deal with the deep searching.